International Journal of Mathematics and Mathematical Sciences 
Volume 2004 (2004), Issue 13, Pages 647-678
doi:10.1155/S0161171204303522

Geometric aspects of high-order eigenvalue problems I. Structures on spaces of boundary conditions

Xifang Cao1 and Hongyou Wu2

 1Department of Mathematics, Yangzhou University, Jiangsu, Yangzhou 225002, China
2Department of Mathematical Sciences, Northern Illinois University, DeKalb 60115, IL, USA
Received 11 March 2002; Revised 3 November 2003

Abstract

We consider some geometric aspects of regular eigenvalue problems of an arbitrary order. First, we clarify a natural geometric structure on the space of boundary conditions. This structure is the base for studying the dependence of eigenvalues on the boundary condition involved, and reveals new properties of these eigenvalues. Then, we solve the selfadjointness condition explicitly and obtain a manifold structure on the space of selfadjoint boundary conditions and several other consequences. Moreover, we give complete characterizations of several subsets of boundary conditions such as the set of all complex boundary conditions having a given complex number as an eigenvalue, and describe some of them topologically. The shapes of some of these subsets are shown to be independent of the quasidifferential equation in question.